The bigraded homotopy groups π, X of a pointed simplicial space X

Abstract

Simplicial spaces are analogues in the category of spaces of chain complexes (i.e., resolutions) in homological algebra. We study a novel way of doing “homotopy theory” with these objects, and describe the associated homotopy groups (which are graded by two indices). These homotopy groups give, for instance, a simple geometric interpretation of the filtration on the homotopy of the realization of a simplicial space provided by the usual homotopy spectral sequence.